A novel approach to axiomatic decision theory

Off-topic, but I cannot resist sharing Peterson's story of how he became interested in decision theory:

A while back a beautiful woman, whom I quite liked, asked me to marry her. I was stunned. Marriage? Now? It is too early! I have not even turned forty! However, for one reason or another I decided not to share my spontaneous reaction with her. I said I felt overwhelmed by this… unexpected proposal, and that I need some time ot think it over. At dawn the following day I… raced to the university library. I borrowed all the books I could find on decision theory. Later the same afternoon, after having learned what modern decision theory is all about, I still had no clue how to answer the lady.

I spent quite a few hours going through Peterson's 2008 book (online copy available here) to see if there were any interesting ideas, and found the time largely wasted. (This was my initial intuition, but I thought I'd take a closer look since Luke emailed me directly to ask me to comment.) It would take even more time to write up a good critique, so I'll just point out the most glaring problem: Peterson's proposal for how to derive a utility function from one's subjective uncertainty about one's own choices, as illustrated in this example:

This means that if the probability that you choose salmon is 2/3, and the probability that you choose tuna is 1/3, then your utility of salmon is twice as high as that of tuna.

What if we apply this idea to the choice between $20 and $30?

Let us now return to the problem of perfect discrimination mentioned above. As explained by Luce, the problem is that ‘the [utility] scale is defined only over a set having no pairwise perfect discriminations, which is probably only a small portion of any dimension we might wish to scale That is, the problem lies in the assumption that p(x  > y) != 0,1 for all x,y in B. After all, this condition is rather unlikely to be satisfied, because most agents know for sure that they prefer $40 to $20, and $30 to $20, etc.

Peterson tries to solve this problem in section 5.3, but his solution makes no sense. From page 90:

Suppose, for example, that I wish to determine my utility of $20, $30, and $40, respectively. In this case, the non-perfect object can be a photo of my beloved cat Carla, who died when I was fourteen. If offered a choice between $20 and the photo, the probability is 1/4 that I would choose the money; if offered a choice between $30 and the photo, the probability is 2/4 that I would choose the money; and if offered a choice between $40 and the photo, the probability is 3/4 that I would choose the money. This information is sufficient for constructing a single ratio scale for all four objects. Here is how to do it: The point of departure is the three local scales, which have one common element, the photo of Carla. The utility of the photo is the same in all three pairwise choices. Let u(photo) = 1. Then the utility of money is calculated by calibrating the three local scales such that u(photo) = 1 in all of them.

So we end up with u($20)=1/3, u($30)=u(photo)=1, u($40)=3. But this utility function now implies that given a choice between $20 and $30, you'd choose $20 with probability 1/4, and $30 with probability 3/4, contradicting the initial assumption that you'd choose $30 with certainty. I have no idea how Peterson failed to notice this.

Prediction from reading this post before I delved into the paper: the controversy is going to be about psychology, not decision theory. (After delving into the paper: I'm going to go with 'prediction confirmed.')

So, he uses six axioms. How do they map onto Howard's 5 that I used? It looks like his 0 is essentially "you can state the problem," his 1 and 2 are my choice, his 3 and 4 don't seem to have mirrors, and his 5 is my equivalence. I find it a little worrisome that three of my axioms don't appear to show up in his- probability, order, and substitution- except possibly in 0. They're clearly present in his analysis, but they feel like things that should be axioms instead of just taken for granted, and it's not clear to me why he needed to raise 3 and 4 to the level of axioms.

It's also not clear to me why he puts such emphasis on the independence and "sure-thing" principle. The "sure-thing" principle is widely held to only apply to a certain class of utility functions / "sure things", and there's not a good reason to expect people should or do have those utility functions. (Outcomes, properly understood, are the entire future- and so a game in which I flip a coin and you win or lose $100 can be different from a game in which I flip a coin and give you either $0 or $200, because you're $100 richer in the second game.) Similarly, independence only holds for normatively correct probability functions, and so if you allow normatively incorrect probability functions you have to throw out independence.

(The difference between independence and "sure-thing" is that "sure-thing" applies a transformation to all of the outcomes, which may not be the same transformation to all of the utilities of those outcomes, and independence applies a transformation to all of the probabilities, which will be the same transformation to all of the act utilities for normatively correct probability functions.)

For the controversy, I'll quote him directly:

I am aware of two objections to the proposed ex ante axiomatization. The point of departure of the first objection is the reasonable suspicion that people simply do not have numerical utilities in their heads, or at least have no access to them. Therefore, to assume that utility is an extensive structure is unreasonable since it seems practically impossible for decision makers to state the preferences required for obtaining the utility function. The same holds true for the probability function; where does the exogenously defined probability function come from?

The probability complaint is uninteresting. The "how do we measure utilities?" complaint is serious but a little involved to discuss.

Basically, there are three branches of rationality: epistemic, instrumental, and terminal (I'm not sure I like "terminal rationality" as a name; please suggest alternatives): thinking about uncertainties and probabilities, thinking about actions, and thinking about outcomes.

Peterson's root complaint is that traditional decision theory is silent on the valuable parts of terminal rationality- it only ensures that your elicited preferences are consistent and then uses them. If they're consistent but insane, the expected utility maximization won't throw up any error flags (for example, Danzig's diet optimization which prescribed 500 gallons of vinegar a day), because checking for sanity is not its job.

But pointing that complaint at decision theory seems mistaken, because it's a question of how you build the utility function. The traditional approach he describes in lukeprog's post above (the woman considering divorce) uses casuistry, expecting that people can elicit their preferences well about individual cases and then extrapolate. I think the approach he prefers is deconstruction- isolate the different desires that are relevant to the outcomes in question, construct (potentially nonlinear) tradeoffs between them, then order the outcomes, then figure out the optimal action. The first checks primarily for internal consistency; the second checks primarily for reflective equilibrium. But both can do each, and they can be used as complements instead of substitutes.

TL;DR: There does not appear to be meat to the controversy over axioms, and if there is then Peterson's axioms strike me as worse than Howard's and possibly worse than vNM or Savage. There is meat to the controversy over discovering utility functions, but I don't think the 2004 paper you link is a valuable addition to that controversy. Compare it to the chapter on choice under certainty from Thinking and Deciding.

Peterson's way of formally representing a decision problem also seems more helpful to me than the ways proposed by Jeffrey and Savage. Peterson (2008) explains:

I shall conceive of a formal representation as an ordered quadruple π =<
A,S,P,U>. The intended interpretation of its elements is as follows.

A = {a1,a2, . . .} is a non-empty set of acts.
S ={s1, s2, . . .} is a non-empty set of states.
P = {p1 : A×S→[0,1], p2 : A×S→[0,1], . . .} is a set of probability functions.
U = {u1 : A×S→Re,u2 : A×S→Re, . . .} is a set of utility functions.

An act is, intuitively speaking, an uncertain prospect. I shall use the term ‘uncertain prospect’ when I speak of alternatives in a loose sense. The term ‘act’ is used when more precision is required.

By introducing the quadruple
, a substantial assumption is made. The assumption implies that everything that matters in rational decision making can be represented by the four sets A,S,P,U. Hence, nothing except acts, states, and numerical representations of partial beliefs and desires is allowed to be of any relevance. This is a standard assumption in decision theory.

A formal decision problem under risk is a quadruple π =
 in which each set P and U has exactly one element. A formal decision problem under uncertainty is a quadruple π =
A,S,P,U in which P = /0 and U has exactly one element. Since P and U are sets of functions rather than single functions, agents are allowed to consider several alternative probability and utility measures in a formal decision problem. This set-up can thus model what is sometimes referred to as ‘epistemic risks’, i.e. cases in which there are several alternative probability and utility functions that describe a given situation. Note that the concept of an ‘outcome’ or ‘consequence’ of an act is not explicitly employed in a formal decision problem. Instead, utilities are assigned to ordered pairs of acts and states. Also note that it is not taken for granted that the elements in A and S (which may be either finite or countable infinite sets) must be jointly exhaustive and mutually exclusive. To determine whether such requirements ought to be levied or not is a normative issue that will be analysed in greater detail in subsequent chapters.

It is easy to jump to the conclusion that the framework employed here presupposes that the probability and utility functions have been derived ex ante (i.e. before any preferences over alternatives have been stated), rather than ex post (i.e. after the agent has stated his preferences over the available acts). However, the sets P and U are allowed to be empty at the beginning of a representation process, and can be successively expanded with functions obtained at a later stage.

The formal set-up outlined above is one of many alternatives. In Savage’s theory, the fundamental elements of a formal decision problem are taken to be a set S of states of the world and a set F of consequences. Acts are then defined as functions from S to F. No probability or utility functions are included in the formal decision problem. These are instead derived ‘within’ the theory by using the agent’s preferences over uncertain prospects (that is, risky acts).

Jeffrey’s theory is, in contrast to Savage’s, homogenous in the sense that all elements of a formal decision problem—e.g. acts, outcomes, probabilities, and utilities—are defined on the same set of entities, namely a set of propositions. For instance, ‘[a]n act is . . . a proposition which it is within the agent’s power to make true if he pleases’, and to hold it probable that it will rain tomorrow is ‘to have a particular attitude toward the proposition that it will rain tomorrow’. In line with this, the conjunction B∧C of the propositions B and C is interpreted as the set-theoretic intersection of the possible worlds in which B and C are true, and so on. Note that Jeffrey’s way of conceiving an act implies that all consequences of an act in a decision problem under certainty are acts themselves, since the agent can make those propositions true if he pleases. Therefore, the distinction between acts on the one hand and consequences on the other cannot be upheld in his terminology, which appears to be a drawback.

However, irrespective of this, the homogenous character of Jeffrey’s set-up is no decisive reason for preferring it to Savage’s, since the latter can easily be reconstructed as a homogenous theory by widening the concept of a state. The consequence of having a six-egg omelette can, for example, be conceived as a state in which the agent enjoys a six-egg omelette; acts can then be defined as functions from states to states. A similar manoeuvre can, mutatis mutandis, be carried out for the quadruple <
A,S,P,U>.

I think it is more reasonable to take states of the world, rather than propositions, to be the basic building blocks of formal decision problems. States are what ultimately matter for agents, and states are less opaque from a metaphysical point of view. Propositions have no spatio-temporal location, and one cannot get into direct acquaintance with them. Arguably, the main reason for preferring Jeffrey’s set-up would be that things then become more convenient from a technical point of view, since it is easy to perform logical operations on propositions. In my humble opinion, however, technical convenience is not the right kind of reason for making metaphysical choices.

An additional reason for conceiving of a formal decision problem as a quadruple <
A,S,P,U>, rather than in the way proposed by Savage or Jeffrey, is that it is neutral with regard to the controversy over causal and evidential decision theory. To take a stand on that issue would be beyond the scope of the present work. In Savage’s theory, which has been claimed to be ‘the leading example of a causal decision theory’, it is explicitly assumed that states are probabilistically independent of acts, since acts are conceived of as functions from states to consequences. This requirement makes sense only if one thinks that agents should take beliefs about causal relations into account: Acts and states are, in this type of theory, two independent entities that together cause outcomes. In an evidential theory such as Jeffrey’s, the probability of a consequence is allowed to be affected by what act is chosen; the agent’s beliefs about causal relations play no role. Evidential and causal decision theories come to different conclusions in Newcomb-style problems. For a realistic example, consider the smoking-caused-by-genetic-defect problem: Suppose that there is some genetic defect that is known to cause both lung cancer and the drive to smoke. In this case, the fact that 80 percent of all smokers suffer from lung cancer should not prevent a causal decision theorist from starting to smoke, since (i) one either has that genetic defect or not, and (ii) there is a small enjoyment associated with smoking, and (iii) the probability of lung cancer is not affected by one’s choice. An evidential decision theorist would, on the contrary, conclude (incorrectly) that if you start to smoke, there is an 80 percent risk that you will contract lung cancer.

It seems obvious that causal decision theory, but not its evidential rival (in its most na¨ıve version), comes to the right conclusion in the smoking-caused-bygenetic-defect problem. Some authors have proposed that for precisely this reason, evidential decision theory should be interpreted as a theory of valuation rather than as a theory of decision. A theory of valuation ranks a set of alternative acts with regard to how good or bad they are in some relevant sense, but it does not prescribe any acts. Valuation should, arguably, be ultimately linked to decision, but there is no conceptual mistake involved in separating the two questions.

A significant advantage of e.g. Jeffrey’s evidential theory, both when interpreted as a theory of decision and as a theory of valuation, is that it does not require that we understand what causality is. The concept of causality plays no role in this theory.

Since there are significant pros and cons for either the causal or evidential approach, it seems reasonable to opt for a set-up that allows for both kinds of theories, until the dispute has been resolved...

The objection to the standard approach is put nicely in chapter 10 of Peterson (2009):

Strictly speaking, when Bayesians claim that rational decision makers behave 'as if' they act from a subjective probability function and a utility function, and maximise subjective expected utility, this is merely meant to prove that the agent's preferences over alternative acts can be described by a representation theorem and a corresponding uniqueness theorem... Bayesians do not claim that the probability and utility functions constitute genuine reasons for choosing one alternative over another. This seems to indicate that Bayesian decision theory cannot offer decision makers any genuine action guidance...

Briefly put, the objection holds that Bayesians 'put the card before the horse' from the point of view of the deliberating agent. A decision maker who is able to state a complete preference ordering over uncertain prospects, as required by Bayesian theories, already knows what to do. Therefore, Bayesian decision makers do not get any new, action-guiding information from their theories.

...the output of a decision theory based on the Bayesian approach is a (set of) probability and utility function(s) that can be used to describe [a rational agent] as an expected utility maximizer... What Bayesians use as input data to their theories is exactly what a decision theorist would like to obtain as output.

...a Bayesian decision theory has strictly speaking nothing to tell [an agent] about how to behave. However, despite this, Bayesian frequently argue that are representation theorem and its corresponding uniqueness theorem are normatively relevant for a non-ideal agent in indirect ways. Suppose, for instance, that [an agent] has access to some of his preferences over uncertain prospects, but not all, and also assume that he has partial information about his utility and probability functions. Then, the Bayesian representation theorems can, it is sometimes suggested, but put to work to 'fill the missing gaps' of a preference ordering, utility function and probability function, by using the initially incomplete information to reason back and forth, thereby making the preference ordering and the functions less incomplete. In this process, some preferences for risky acts might be found to be inconsistent with the initial preference ordering, and for this reason be ruled out as illegitimate.

That said, this indirect use of Bayesian decision theory seems to suffer from at least two weaknesses. First, it is perhaps too optimistic to assume that the decision maker's initial information always is sufficiently rich to allow him to fill all the gaps in his preference ordering... The second weakness is that even if the initial information happens to be sufficiently rich to fill the gaps, this manouevre offers no theoretical justification for the initial preference ordering over risky acts. Why should the initial preferences be retained? ...As long as no preference violates the structural constraints stipulated by the Bayesian theory, everything is fine. But is not this view of practical rationality a bit too uninformative?

I don't currently view the two objections in the second-to-last paragraph above as necessitating a non-Bayesian decision theory, but the original concern that Bayesian decision theory doesn't technically offer any action-guidance is serious, and the primary motivation for my interest in Peterson's ex ante approach.

The final page of Peterson (2009) is also quotable:

Surely people could learn more about decision theory and eventually become more rational! In fact, this textbook is part of an attempt to make that happen.